Mathematical Modeling

Mathematical modeling of dynamic system

Mathematical modeling is essential before any type of control system in world for more articles visit these articles.

Dynamic system is a system whose states change with reference to an independent variable such as time.

Examples

  • Moving car, states are the position, velocity, heading direction.
  • Electric power generator, states are terminal voltage, frequency of the voltage wave form.
  • Air conditioning system, states are the temperature, humidity in the indoor system

If the states are not controlled, they may end up at anywhere, this can be highly unacceptable and sometimes may lead to hazardous situation.

Mathematical model of dynamic system essentially are differential equations.

Examples

  • Electrical circuits

VL(t)= L

Ic(t)= L

Hence Kirchhoff voltage equation yield differential equations.

  • Mechanical systems

Fs(t) = kx(t)

Fd(t)= D(dX(t)/dt)

F(t)= m(dX2/dt)

  • Thermal systems

Covered by laws of thermodynamics recall newtons law of cooling for instance

Dynamic systems we consider are electrical, electromagnetic, mechanical, electromechanical, chemical and thermal or variety of combinations of them.

Laplace Transform for mathematical modeling

Laplace transform is a mathematical tool that can use to solve linear differential equations. The Laplace transform of a time domain function or signal f(t) is defined as

Laplace operator
Laplace operator
mathematical modelling  Laplace conversion
Laplace conversion

M = mass

FE = Engines? Breaking torque

D = damping constant

V = velocity

F = ma                                                             (Assume zero initial conditions)

mathematical model equation  mathematical modelling
mathematical model equation

GP(S) is call the “transfer function”

FE(t)     Car    V(t) converts to FE(S)    Car V(S)

However, if we interest in position, u(t) of the car

mathematical model equation
mathematical model equation

By taking Laplace transform and assuming zero initial condition, X(0) =0

V(S) – SX(S) = 0

From equation (1)

mathematical model equation
mathematical model equation

If we want to understand how the current, i(t) in the circuit may be control by adjusting the supply voltage V(t) as the input, by using KVL

mathematical model equation
mathematical model equation

By taking Laplace transform and assuming zero initial conditions i(0) = 0

mathematical model equation
mathematical model equation

I(S) – Laplace input functions

E(S) is Laplace output functions

Control System in Laplace Domain

E(S) = R(S) – H(S)C(S)

GC(S)GP(S)E(S) = C(S)

Closed loop Function
Closed loop Function

m(S) is closed loop transfer function

Gc(S) is controller transfer function

Gp(S) is plant transfer function

H(S) is feedback element transfer function

C(S) is output transfer function

R(S) is reference input transfer function

Controller design explained

Control objective

e(t) ——-> 0; E(S) ——-> 0

E(S) = R(S) – C(S)H(S) ——-> R(S) = C(S)H(S)

Closed loop Function
Closed loop Function

Stability of a control system

Stabilizing the control system must do prior to controller design and tuning. It is impossible to work with an unstable system. For instance, when you are learning to ride a bicycle, first step is to learn to stay stable (without falling or getting out of the road). Learning fine skills come later. Stability refers to firmness or steadiness, once the system has been stabilized optimizing the control performance is then possible.

Routh’s Stability criterion for mathematical modeling

If the transfer function of the system known, stability of linear systems can check using Routh’s criterion. Transfer function defined using familiar notation as

 1 + Gc(S)Gp(S)H(S) = 0

This is usually a polynomial function of S that may written in the following form

Mathematical modeling routh criteria
routh criteria

Now, Construct the Routh’s array as follow

Mathematical modeling routh criteria
routh criteria

While

Mathematical modeling routh criteria
routh criteria

Condition for stability for mathematical modeling

Provided  > 0, all the elements in the first column must be non negative.

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    Een andere gebruiker van lorazepam heeft ook een hele opluchting ervaren van het gebruik van deze specifieke voorgeschreven medicatie.

    Een persoon gebruikte deze medicatie als een manier om te kalmeren bij het kijken naar een film, het sporten, het maken van wandelingen of andere activiteiten hielp niet.

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    Succespercentages
    Ativan heeft een succespercentage van 85%. Het is ongeveer 88%
    effectief voor het wegwerken van angstsymptomen en paniekaanvallen.
    Het is ongeveer 83% effectief voor het verwijderen van bipolaire stoornis en alcoholisme symptomen. Dit product heeft ook een 68% effectiviteitspercentage
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